How is AlphaZero different from Stockfish or Rybka?
Stockfish is free software (source while AlphaGo can only be used by employees at Deepmind. This is important, because it means all answers to this question can only rely on the AlphaGo paper. Some decisions during the matches (e.g. giving limited time per move and not per game; giving different computational power) lead to unclarity how good alpha go really is.

Stockfish  uses Alpha-beta search with a hardcoded metric to evaluate how good a position is.

In contrast, AlphaGo uses Monte Carlo tree search and convolutional neural networks. They are trained in an reinforcement learning setting.

AlphaGo is promoted as being almost directly applicable to different settings (Go, Chess), while stockfish is only applied to Chess. The techniques of stockfish can also be applied to Go, but the problem is the board evaluation. For Chess it's easy compared to Go.

References


Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm, December 2017.
Mastering the Game of Go without Human Knowledge is about Alpha Go Zero